连续介质力学中力偶和偶应力的性质
Character of couple and couple-stress in continuum mechanics
摘要：
本文从基本的角度考察了力学中力矩和力偶的概念。结果表明，虽然用力矩矢量表示力偶在刚体力学和材料强度中是非常有用的，但在连续介质力学中却有很大的误导性。在连续介质力学中，要确定集中力偶的作用，不仅要确定力偶矩，而且要确定其组成部分平行反力的作用线。然而，在连续体的平衡或运动的控制方程中，只有力偶矩、力偶牵引矩和力偶应力矩出现，而没有指定任何力偶密度力的作用线。这导致了连续介质中应力和变形状态的非唯一性，这表现在偶应力张量的不确定性上。然而，应力和变形在连续介质中的物理状态是唯一的和确定的。因此，这就对体力偶、力偶-牵引和力偶-应力的形式产生了一定的限制。在这里，在连续体中的相互作用的唯一性被用来建立连续体不支持体力偶的分布，以及具有法向力矩的表面扭转力偶-牵引力的分布。由此还建立了力偶牵引作为剪力-牵引双层的作用机制，以及连续介质力学中力偶应力矩张量的斜对称性质。
英文摘要：
In this paper, the concept of moment and couple in mechanics is examined from a fundamental perspective. It turns out that although representing a couple by its moment vector is very useful in rigid body mechanics and strength of materials, it has been very misleading in continuum mechanics. To specify the effect of a concentrated couple in continuum mechanics, not only the couple moment, but also the line of action of its constituent parallel opposite forces must be specified. However, in the governing equations of equilibrium or motion of a continuum, only moment of body couple, moment of couple-tractions, and moment of couple-stresses appear without specifying the line of action of any couple density forces. This results in non-uniqueness of the state of stresses and deformation in the continuum, which has shown itself in the indeterminacy of the couple-stress tensor. Nevertheless, the physical state of stress and deformation in the continuum is unique and determinate. Therefore, this imposes some restrictions on the form of body couple, couple-traction and couple-stresses. Here, the uniqueness of interactions in the continuum is used to establish that the continuum does not support a distribution of body couple, and a distribution of surface twisting couple-traction with normal moment. From this the mechanism of action of the couple-traction as a double layer of shear force-tractions, and the skew-symmetric character of the couple-stress moment tensor in continuum mechanics are also established.
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